Accessibility of the critical point in a Cremer Julia set

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a polynomial $f$ of degree at least two, $f^n$ is its $n$-fold iterate, $K(f)=\{z\in\mathbb C:(f^n(z))_{n\geq0}\text{ is bounded}\}$ is its filled Julia set, and $J(f)=\partial K(f)$ is its Julia set. A space is locally connected if each point has a neighbourhood basis of connected sets. In Milnor's terminology here, a Cremer polynomial is $P_\alpha(z)=z^2+\lambda z$, where $\lambda=e^{2\pi i\alpha}$, $\alpha\in\mathbb R\setminus\mathbb Q$, and no holomorphic change of coordinate near $0$ conjugates $P_\alpha$ to $w\mapsto\lambda w$. Its unique critical point is $-\lambda/2$. A point $x\in J(f)$ is accessible from $\mathbb C\setminus J(f)$ if there is a continuous curve $\eta:[0,1]\to\mathbb C$ with $\eta([0,1))\subset\mathbb C\setminus J(f)$ and $\eta(1)=x$.

Statement recorded in the supplied source.

Can the critical point $-\lambda/2$ of a Cremer polynomial $P_\alpha$ be accessible from $\mathbb C\setminus J(P_\alpha)$? [S2]

Short English statement

Can a curve outside a Cremer quadratic Julia set reach its critical point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300030 (AMR-052-0030). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Milnor, "Problems on local connectivity", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 5, pp. 21; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.